Concerning the Design of Capacitively Driven Induction Coil Guns

IEEE TRANSACTIONS ON PLASMA SCIENCE. VOL. 17. NO. 3. JUNF 19x9 429 Concerning the Design of Capacitively Driven Induction Coil Guns Abstract-This paper is concerned with the design of capacitively 11. THEORYAND SIMULATIONMODEL driven, multisection, electromagnetic coil launchers, or coil guns, tak- A. Capacitively Driven Coil-huncher System ing their transient behavior into account. A computer simulation based on the viewpoint of lumped parameter circuits is developed to predict Fig. 1 is a sketch of a capactively driven, multisection the performance of the launcher system. It is shown that a traveling coil launcher which consists of two main parts: The barrel electromagnetic wave can be generated on the barrel by the resonance and the projectile. It may be driven either by a set of ca- of drive coils and their capacitors. More than half of the energy ini- pacitors, as indicated in the diagram, or, alternatively, by tially stored in the capacitor bank can be converted into kinetic energy a set of ac polyphase sources. The barrel is a linear array of the projectile in one shot, and an additional quarter can be utilized in subsequent shots, if the launcher dimensions, resonant frequency, of drive coils. The projectile is a conducting sleeve sur- and firing sequence are properly selected. The projectile starts smoothly rounding the payload. The drive coils on the barrel are from zero initial velocity and with zero initial sleeve current. Section- energized with a certain time sequence so as to generate to-section transitions which have significant effects on the launcher a traveling electromagnetic wave that interacts with a sys- performance are also discussed. Experimental results were obtained tem of currents in the sleeve. with a small model and are in good agreement with theoretical predictions. Two modes of operation are possible for the launcher: The “synchronous” mode and the “asynchronous” mode. In the first case, the velocity of the projectile is I. INTRODUCTION exactly equal to that of the traveling wave. For this mode a set of currents must be impressed on the sleeve; each N advantage that electromagnetic launchers have over coil must be energized separately; the timing of the clo- A chemical ones is that they can accelerate projectiles sure of each switch must be governed by the position of to much higher (hyper) velocities. The present interest in the projectile; and the size of each capacitor must be cho- electromagnetic launchers has resulted in a great compe- sen in such a way that the frequency matches the dM/dx tition between the two major types: Rail guns and coil curves of the drive coils. Because of the complexity of guns. Although rail guns are currently at a more advanced the control system and the power conditioner associated stage of development, coil guns have their own desirable with this mode, we discuss instead in detail here the asyn- characteristics which have been discussed in the literature chronous mode in which the projectile velocity is smaller [1]-[5]. The operating principles of various kinds of coil than that of the traveling wave so that the relative motion launchers have been summarized by Mongeau et al. [2]. between the two causes induced currents in the projectile. The performance of such launchers is still being investi- In this case the coils may be connected in series or in gated. In a previous paper [4], design principles based on parallel to form a limited number of phase windings, as the assumption that a steady state prevails in each section is done in conventional machines. However, since the of the gun were outlined. Here, instead, account is taken losses in the sleeve are proportional to its “slip” behind of the fact that the steady state is actually never reached the traveling wave, and since, in a launcher, the projectile during the transit of the projectile through the gun. velocity ranges over at least one order of magnitude, it The first part of this paper describes the physical ar- becomes necessary to increase the velocity of the travel- rangement and the computer simulation model. The sec- ing wave stepwise by dividing the barrel into sections. In ond part discusses the design. Numerical results and per- order to obtain a wave whose phase velocity increases formance estimates for a launcher now under construction from section to section along the length of the barrel, are given in the third part, which contains also a compar- either the frequency for the pole pitch (half wavelength) ison between theoretical predictions and experimental data must increase. Because of the relatively short length of obtained with a small laboratory model. the projectile it is not practical to increase the pitch ade- quately. Therefore the resonant frequency must increase Manuscript received October 14, 1988; revised February 8, 1989. This along the length of the barrel and different capacitor sizes work was supported by the SDIO/IST and managed by the AFOSR under must be used in different sections. Contract no. F49620-86-C-0126 and by the USASDC under Contract no. DASG60-88-C-0047. B. System Equations The authors are with the Polytechnic University, 333 Jay Street, Brook- lyn, NY 11201. The coil launcher of Fig. 1 is modeled by the equivalent IEEE Log Number 8927934. circuit shown in Fig. 2. Because the axial distribution of 0093-3813/89/0600-0429$01.OO 0 1989 IEEE 430 IEEE TRANSACTIONS ON PLASMA SCIENCE. Vol.. 17. NO. 3. JUNE IYXY where [ C 3 is a diagonal (n element) matrix containing the capacitances and [ V,] and [Id ] are the capacitor voltage and the drive-coil current column matrices, respectively. [ V,] and [Id ] are submatrices of [ VI and [I3. The Lorentz force acting on the projectile is given by F = 1 [I]TIG][I] (3) where [ G 3 = d [ M ] /dx and the superscript T stands for Fig. l. Sketch of multisection capacitively driven coil launcher. the transpose of the matrix. Finally, we consider the equa- tion of motion and combine it with (1)-(3). The complete set of equations which describe the capacitively driven coil-launcher system is then given in matrix form: d { [L] + [MI} ;[I1 = [VI - [RI[I] - vp[G][I] (m + n eqns.) (4) d V IVcl = -[Id] (n eqns.) (5) + [cl Fig. 2. Lumped parameter circuit model of the capacitively driven coil launcher. the induced current in the sleeve is not uniform, to sim- ulate the physical system we divide the sleeve into several dx - = up (1 eqn.) zones that are electrically insulated from each other and dt represent them as separate projectile coils. The number of coils into which we divide the sleeve depends on the re- where Mp is the mass of projectile and up is the velocity quirement of accuracy. The number of drive coils de- of the projectile. Equations (4)-(7) represent a set of pends on performance specifications such as the muzzle simultaneous nonlinear differential equations with time- velocity, weight of the projectile, the caliber, and the variable coefficients. The number of first-order differen- length of the barrel. We denote by n the number of drive tial equations involved in the system is coils, and by m the number of projectile coils. Employing Kirchhoff’s voltage law, we write the net- N = 2n + m + 2 (8) work equations in matrix form: in 2n + m + 2 variables (mprojectile currents, n drive currents, n driver capacitor voltages, up, and x). [VI = [Rl[Il + { [LI[Il + [MI[Il) (1) f C. Analysis of the System Energies where [ VI and [I]are column (m + n element) matrices To assess the performance of the launcher system one composed of the individual projectile and drive-coil volt- needs first to consider the energy balance relations. This ages and currents, respectively. [ R ] and [ L] are diagonal will eventually lead to the optimum design of the launcher. (m + n element) matrices composed of the individual Basically, the energy balance involves several items: The projectile and drive-coil resistances and self-inductances, electric energy stored in the capacitor bank, the magnetic respectively, and [MIis a square (m + n) X (m + n) energy stored in the coils, the kinetic energy gained by matrix, each element of which represents a mutual induc- the projectile, and the ohmic loss due to the resistances. tance between any two individual coils. The formulas used The main goal in the design of a capacitively driven coil to evaluate each element of [ L]and [ M ] are given in [6] launcher is to achieve the transfer of as much as possible and [7]. It should be noted that the mutual inductances of the electric energy stored in the capacitors into kinetic [ M ] between drive coils and projectile coils are functions energy of the projectile to attain high system efficiency. of the distance x which denotes the position of the projec- The energy relations are derived as follows below. tile and is a function of time. This complicates matters Multiplying (1) by [I] and rewriting it in an alterna- considerably. tive form, we obtain: The relations between the capacitor voltages and drive- coil currents are HE et 01.: CONCERNING THE DESIGN OF CAPACITIVELY DRIVEN INDUCTION COIL GUNS 43 1 Since This average acceleration, as calculated by the simulation code, is an important quantity for the design of the launcher because it affects the dimensions of the launcher. Another key quantity is the Energy Transfer Ratio (ETR), d IT which is defined as the ratio of the gain in kinetic energy = [IIT([L] + [MI) [I1 -k 5 {[I] ~P[GI[I]) to the total energy initially stored in the capacitors, because it affects the weight of the capacitor bank: where wkf and wk, refer to the final and initial kinetic energies of the projectile respectively, Wcap(0) stands for initial capacity energy, uj and U, are the final and initial We note in (1 1) that the term on the left represents the velocities of the projectile, and vcd(0)is the initial ca- electric power in the capacitor.

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